-
PERMUTATION RESOLUTIONS
AND COHOMOLOGICAL SINGULARITY
PAUL BALMER AND MARTIN GALLAUER
Abstract. We consider the bounded derived category of finitely
generated
representations of a finite group, with coefficients in a
noetherian commutativering. When the coefficient ring in question
is regular we prove that this derived
category can be built from a simpler one, namely the homotopy
category of
permutation modules, via localization and idempotent-completion.
When thecoefficient ring is singular, we construct an invariant to
decide when a complex
reduces to permutation modules. This invariant uses group
cohomology and
takes values in the big singularity category of the coefficient
ring.
Contents
1. Introduction 12. Bounded permutation resolutions 53.
Essential image of the functor F̄ 94. Big singularity category 135.
Cohomological singularity 156. Main result 187. Localization of big
categories 228. Density and Grothendieck group 24References 28
1. Introduction
In the whole paper, G is a finite group and R is a commutative
noetherian ring.
In this generality, the group algebra RG combines the complexity
of commutativealgebra and of representation theory. Even ignoring
the G-action, one does notexpect a simple description of finitely
generated R-modules. And even when wehave one, say when R is a
field, if that field has characteristic p > 0 dividing theorder
of G, we still face the wild complexity of modular representation
theory.
Much tidier is the class of permutation RG-modules, i.e. those
isomorphic to a freeR-module R(A), with basis a finite G-set A and
with G-action R-linearly extendedfrom the one on A. Any such module
is isomorphic to R(G/H1)⊕ · · · ⊕ R(G/Hr)for a choice of H1, . . .
,Hr among the finitely many subgroups of G.
Date: September 18, 2020.Key words and phrases. Modular
representation theory, permutation modules, singularity cat-
egory, group cohomology, derived categories.First-named author
supported by NSF grant DMS-1901696. The authors would like to
thank
the Isaac Newton Institute for Mathematical Sciences for support
and hospitality during theprogramme K-theory, algebraic cycles and
motivic homotopy theory when work on this paper was
undertaken. This programme was supported by EPSRC grant number
EP/R014604/1.
1
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2 PAUL BALMER AND MARTIN GALLAUER
Despite their apparent simplicity, permutation modules play a
non-trivial role inmany areas of group representation theory, as
recalled for instance in the introduc-tion of Benson-Carlson
[BC20]. Our own interest in permutation modules stemsfrom their
connection with Artin motives, following Voevodsky [Voe00, §
3.4].
Let us consider the inclusion of the additive category of
permutation RG-modulesinto the abelian category of finitely
generated RG-modules, denoted as follows:
(1.1) perm(G;R) ⊆ mod(RG)
It gives a canonical functor F : Kb(perm(G;R))→ Db(mod(RG)) from
the boundedhomotopy category of the former to the bounded derived
category of the latter. Thekernel of F is the thick subcategory
Kb,ac(perm(G;R)) of acyclic complexes of per-mutation modules,
studied in [BC20]. The functor F descends to the
correspondingVerdier quotient and, after idempotent-completion
(...)\, yields a functor
(1.2) F̄ :
(Kb(perm(G;R))
Kb,ac(perm(G;R))
)\−→Db(mod(RG)).
This canonical functor F̄ is our main object of study. The only
formal propertythat F̄ inherits by construction is being
conservative. So the first surprise is:
1.3. Theorem. The canonical functor F̄ of (1.2) is always fully
faithful.
This is part of our main result that furthermore identifies the
image of F̄ . Foran object X in Db(mod(RG)) to be in the image of
F̄ , it is clearly necessary for its
underlying complex of R-modules ResG1 (X) to be perfect over R.
For G = 1, thisis obviously the only condition. However we prove in
Example 5.16 that this naivenecessary condition is not sufficient
for general groups. To state the correct result,we invoke the
singularity categories of the ring R (which do not depend on G)
Dsingb (R) :=Db(mod(R))
Dperf(R)and Dsing(R) := Kac(Inj(R)).
The construction of the right-hand ‘big’ singularity category
Dsing(R), followingKrause [Kra05], can be found in Recollection
4.1. It is a compactly generatedtriangulated category, whose
subcategory of compact objects coincides with
theidempotent-completion of the left-hand Dsingb (R), the perhaps
more familiar (small)singularity category ; see Orlov [Orl04]. The
name ‘singularity category’ comes from
the fact that R is regular if and only if Dsing(R) = Dsingb (R)
= 0. More generally,
Stevenson [Ste14] relates Dsingb (R) to the singularity locus of
R via tensor-triangular
geometry. Krause also extends the obvious quotient functor
Db(mod(R))�Dsingb (R)
to unbounded complexes. We call this extension the singularity
functor
sing : D(R)→ Dsing(R).
For each subgroup H ≤ G, let (−)hH : D(RG) → D(R) be the
right-derivedfunctor of the H-fixed-points functor (−)H . We can
now state our main result.
1.4. Theorem (Theorem 6.16). The canonical functor F̄ of (1.2)
is fully-faithfuland its essential image consists of those X ∈
Db(mod(RG)) such that the invariant
(1.5) χH(X) := sing(XhH)
vanishes in the big singularity category Dsing(R), for every
subgroup H ≤ G.
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PERMUTATION RESOLUTIONS AND COHOMOLOGICAL SINGULARITY 3
The functor (−)hG : D(RG)→ D(R) is the derived-category version
of ordinarygroup cohomology, that is, the following left-hand
square commutes:
(1.6)
Mod(RG) �
//
H∗(G,−)��
D(RG)
(−)hG
��
χG
(def)%%
Mod(R) D(R)sing
//H∗
oo Dsing(R).
We call the invariant χH : D(RG)ResGH−−−→ D(RH) (−)
hH
−−−−→ D(R) sing−−→ Dsing(R) of (1.5)the H-cohomological
singularity functor, for every subgroup H ≤ G. To apply The-orem
1.4 to an X whose underlying complex ResG1 (X) is perfect it
suffices to testχH(X) = 0 for the Sylow subgroups H of G, or
alternatively for the (maximal)elementary abelian subgroups. In
particular, if G is a p-group, there are two condi-tions for a
complex X to belong to the image of F̄ : the naive ResG1 (X) ∈
Dperf(R)and the new χG(X) = 0 in Dsing(R).
To appreciate the strength of Theorem 1.4, observe that when the
ring R isregular, the condition χH(X) = 0 is trivially true in
Dsing(R) = 0. And yetthis recovers a non-trivial result due to
Rouquier (unpublished; see [BV08, § 2.4]or
https://www.math.ucla.edu/~rouquier/papers/perm.pdf).
1.7. Corollary (Rouquier; see Corollary 6.20). Let R be a
regular commutativering. The canonical functor F̄ of (1.2) is a
triangulated equivalence(
Kb(perm(G;R))
Kb,ac(perm(G;R))
)\∼−−→ Db(mod(RG)).
In fact, we give here two proofs of this fact: one as a
corollary of Theorem 1.4 anda more direct one which does not use
cohomological singularity, in Scholium 3.15.
In the even more special case where R = k is a field, the main
result of Balmer-Benson [BB20] says that every kG-module M admits a
‘complement’ N such thatM ⊕ N possesses a finite resolution by
permutation modules. We give a derived-category reformulation of
this fact in Corollary 3.11, that implies Corollary 1.7 forR = k.
Corollary 3.11 also yields some control on the complement N of
[BB20]in terms of M , using Thomason’s classification [Tho97, § 2]
of ‘dense’ triangulatedsubcategories via Grothendieck groups. As
application, we obtain for instanceCorollary 8.7, whose statement
does not involve derived categories:
1.8. Corollary. Let M be a finitely generated kG-module, where k
is a field.Consider the Heller loop of M defined by Ω(M) := Ker(kG
⊗k M�M). ThenM ⊕Ω(M) admits a finite resolution 0→ Pn → · · · → P0
→M ⊕Ω(M)→ 0 whereall Pi are finitely generated permutation
kG-modules.
Replacing permutation by p-permutation modules (a. k. a. trivial
source modules)we will also prove the following striking result
(Theorem 8.17).
1.9. Theorem. Let k be a field of characteristic p > 0. Every
finitely generatedkG-module admits a finite p-permutation
resolution.
In particular for p-groups the ‘complement’ N of [BB20] can be
taken to be zero:
1.10. Corollary. If the field k has characteristic p > 0 and
G is a p-group thenevery finitely generated kG-module has a finite
resolution by permutation modules.
https://www.math.ucla.edu/~rouquier/papers/perm.pdf
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4 PAUL BALMER AND MARTIN GALLAUER
Finally, let us mention a variant for ‘big’ triangulated
categories. The Neeman-Thomason Theorem [Nee92] suggests that the
equivalence of Corollary 1.7 is thecompact tip of an iceberg
involving compactly generated triangulated categories.Indeed,
Krause [Kra05] proved that Db(mod(RG)) is the category of
compactobjects in the homotopy category of injectives K(Inj(RG)).
So we expect theexistence of a ‘big’ triangulated category
DPerm(G;R) whose compact objectswould consist of Kb(perm(G;R))
\ and such that the finite localization with re-spect to
Kb,ac(perm(G;R)) coincides with K(Inj(RG)). This is what we prove
inCorollary 7.17 for R regular. We construct DPerm(G;R) and
describe the corre-sponding finite localization even when R is
singular. A more thorough expositionwill appear in the companion
paper [BG20], where we also explain the connectionwith
cohomological Mackey functors and with Artin motives in some
detail.
The organization of the paper is as follows. We discuss a
refinement of thecategory of permutation modules in Section 2, that
is technically more convenient.We identify it with the essential
image of F̄ in Section 3. Section 4 is dedicatedto Krause’s big
singularity category and Section 5 discusses our
‘cohomologicalsingularity’ functor (1.5). We prove our main result
in Section 6 and the variantfor big categories in Section 7.
Finally we deduce the corollaries about permutationresolutions in
Section 8. Note that we already prove the field case of our
mainresult in Section 3. So the field-focussed reader can follow
the shortcut Section 2→ Section 3 → Section 8.
Acknowledgements. We thank Robert Boltje and Serge Bouc for
removing our ear-lier assumption that the field k should be
‘sufficiently large’ in Proposition 8.10 andits corollaries.
Notation and convention.We write ' for isomorphisms and reserve
∼= for canonical isomorphisms.A commutative noetherian ring R is
regular if it is locally of finite projective
dimension. Most results reduce to the case where R is connected
(not a product).Unless specified, modules are left modules. We
denote by Mod(?) the category
of modules and by mod(?) the subcategory of finitely generated
ones.Since fixed points (−)H and other decorations (duals) appear
in exponent, we
use homological notation for complexes · · · → Mn → Mn−1 → · · ·
. We write K(and Kb) for homotopy categories of (bounded)
complexes, and D (and Db) for(bounded) derived categories. We
abbreviate Db(RG) for Db(mod(RG)). Whenwe speak of a module M as a
complex, we mean it concentrated in degree zero.
All triangulated subcategories are implicitly assumed to be
replete. We abbre-viate ‘thick’ for ‘triangulated and thick’ (i.e.
closed under direct summands). Atriangulated subcategory is called
localizing if it is closed under coproducts.
We denote by A\ the idempotent-completion (Karoubi envelope) of
an additivecategory A, or its obvious realization in some ambient
idempotent-complete cate-gory. See [BS01] for details, including
the fact that Kb(A
\) ∼= Kb(A)\.
1.11. Recollection. We defined the category perm(G;R) of
permutationRG-modulesbefore (1.1). If G is a p-group and R = k is a
field of characteristic p, this categoryis idempotent-complete
([Fei82, IX, 3.4]). For a general group G, summands ofpermutation
modules are called p-permutation modules, or trivial source
modules.A kG-module is p-permutation if and only if its restriction
to a p-Sylow subgroup is
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PERMUTATION RESOLUTIONS AND COHOMOLOGICAL SINGULARITY 5
a permutation module. This characterization is specific to
fields of characteristic pwhereas the idempotent-completion
perm(G;R)\ makes sense for any ring R.
We use the phrase ‘M is \-permutation’ to say that M belongs to
perm(G;R)\.
2. Bounded permutation resolutions
Until specified otherwise, we assume all RG-modules finitely
generated and allcomplexes bounded. As in [BB20], we begin with a
stronger notion of resolution.
2.1. Definition. Let X be a bounded complex of RG-modules and m
∈ Z. An m-freepermutation resolution of X is a quasi-isomorphism of
complexes s : P → X whereP is a bounded complex of permutation
RG-modules which is m-free, meaning thatPi is free for all i ≤ m.
Clearly m′-free implies m-free when m′ ≥ m.
Similarly (Recollection 1.11) an m-projective \-permutation
resolution is a quasi-isomorphism P → X where all Pi are
\-permutation, and projective for i ≤ m.2.2. Remark. The statements
of Lemma 2.4, Corollary 2.5 and Proposition 2.7below also hold with
the words ‘permutation’ replaced by ‘\-permutation’ and with‘free’
replaced by ‘projective’. We leave most of their proofs to the
reader.
2.3. Remark. The word ‘resolution’ in Definition 2.1 can be
misleading for thecomplex P is allowed to extend further to the
right thanX itself, even forX = M [0],a single RG-module in degree
zero. This can be corrected, when m is large enough:
2.4. Lemma. Let m ≥ n be such that the complex X is acyclic
strictly below degree nand such that X admits an m-free permutation
resolution. Then X admits an m-free resolution P → X where Pi = 0
for all i < n. (See Remark 2.2.)Proof. We can assume n = 0. So m
≥ 0. Let s : Q → X be an m-free permuta-tion (resp. m-projective
\-permutation) resolution. Since s is a quasi-isomorphism,Hi(Q) = 0
for i < 0. Since Qi is projective for i < 0, the complex Q
‘splits’ innegative degrees. So we have a decomposition Q = Q′ ⊕Q′′
where the subcomplexQ′ = · · · → Q1 → Q′0 → 0 → · · · is
concentrated in non-negative degrees whereasQ′′ = · · · → 0 → Q′′0
→ Q−1 → · · · lives in non-positive degrees and is acyclic,i.e. the
composite Q′ → Q→ X remains a quasi-isomorphism. At this stage Q′0
isonly projective but not necessarily free. (In the case of
m-projective \-permutationresolutions, the proof stops here with P
= Q′.) Since Q′′ is split exact, we see thatQ′′0 is stably free:
Q
′′0 ⊕ (⊕i
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6 PAUL BALMER AND MARTIN GALLAUER
Suppose that m ∈ Z is such that Xi = 0 and Yi = 0 for i > m
and Pi is projectivefor all i ≤ m. Then there exists f̂ : P → Y
such that s f̂ is homotopic to f .
Proof. Let Z be an acyclic complex such that Zi = 0 for i >
m+ 1. Then any mapP → Z is null-homotopic, as one can build a
homotopy using the usual inductionargument, that only requires Pi
projective for i ≤ m to lift against the epimorphismZi+1 � im(Zi+1
→ Zi). Now Z := cone(s) is such a complex. So the compositeP
f−→ X → cone(s) is zero in Kb(mod(RG)). Hence f factors as
claimed. �
2.7. Proposition. Let s : Y → X be a quasi-isomorphism. Then X
admits m-freepermutation resolutions for all m ≥ 0 if and only if Y
does. (See Remark 2.2.)
Proof. From Y to X is trivial. So suppose that X has the
property and let us showit for Y . Let m ≥ 0. Increasing m if
necessary, we can assume that Xi = Yi = 0for all i > m. Let then
f : P → X be an m-free permutation resolution of X. ByLemma 2.6,
there exists f̂ : P → Y such that s f̂ ∼ f . By 2-out-of-3, f̂ : P
→ Y isa quasi-isomorphism, hence yields an m-free permutation
resolution of Y . �
In Definition 2.1, we dealt with complexes on the nose. The
above propositionallows us to pass to the derived category, if we
make sure to require the existenceof m-free permutation resolutions
for all m ≥ 0, not just for some specific m.
2.8. Definition. We have well-defined replete subcategories of
the derived category
P(G;R) =
{X ∈ Db(RG)
∣∣∣∣ X admits m-free permutation resolutionsin the sense of
Definition 2.1, for all m ≥ 0},
Q(G;R) =
{X ∈ Db(RG)
∣∣∣∣ X admits m-projective \-permutation resolutionsin the sense
of Definition 2.1, for all m ≥ 0}.
2.9. Proposition. The two subcategories P(G;R) ⊆ Q(G;R) above
are triangulatedsubcategories of Db(RG).
Proof. We prove it for P(G;R). The proof for Q(G;R) is similar.
It suffices toshow that if f : X → Y is a morphism in Db(RG) with
X,Y ∈ P(G;R) thencone(f) ∈ P(G;R). The morphism f is represented by
a fraction X s← Z g−→ Yin Chb(RG) with s a quasi-isomorphism. By
Proposition 2.7, we have Z ∈ P(G;R).Since cone(f) ' cone(g) in
Db(RG), we can assume that f : X → Y is a plainmorphism of
complexes. Let now n ≥ 0. Since Y belongs to P(G;R), choose
ann-free permutation resolution t : Q → Y . Choose m � n such that
Qi = Yi = 0for i > m, using that Q and Y are bounded. Since X ∈
P(G;R), choose an m-freepermutation resolution s : P → X. We thus
have the (plain) morphisms f, s, t:
Ps��
h// Q
t��
Xf// Y
By Lemma 2.6, there exists h : P → Q such that t h ∼ f s. Since
s and t arequasi-isomorphisms, so is the induced map cone(h)→
cone(f). Now, the mappingcone of h is a complex of permutation
modules that is free in degrees ≤ n since Pand Q are. As n ≥ 0 was
arbitrary, we proved cone(f) ∈ P(G;R) as claimed. �
2.10. Example. Let G be a p-group and R = k a field of
characteristic p. ThenDb(kG) is generated as a triangulated
subcategory by k. So, by Proposition 2.9,
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PERMUTATION RESOLUTIONS AND COHOMOLOGICAL SINGULARITY 7
the triangulated subcategory P(G; k) = Q(G; k) is equal to
Db(kG) if and only if itcontains k. We will prove that this holds
in Theorem 8.17.
2.11. Recollection. A triangulated subcategory A ⊆ T is dense if
every object Xof T is a direct summand of an object X ⊕ Y of A.
This amounts to X ⊕ΣX ∈ Asince X ⊕ ΣX = cone
(( 0 00 1 ) : X ⊕ Y → X ⊕ Y
). Hence the thick closure of A is
thick(A) ={X ∈ T
∣∣∃Y ∈ T s.t. X ⊕ Y ∈ A} = {X ∈ T ∣∣X ⊕ ΣX ∈ A}.When T is
idempotent-complete (like here Db(RG)) we have thick(A) = A
\.
2.12. Proposition. The triangulated subcategory P(G;R) is dense
in Q(G;R).
Proof. It suffices to show that for every X ∈ Q(G;R), we have X
⊕ΣX ∈ P(G;R).By Definition 2.1, it suffices to show that if P is an
m-projective complex of \-permutation RG-modules for some m ≥ 0
then P⊕ΣP is homotopy equivalent to anm-free complex P̃ of
permutation RG-modules. For each i, since Pi is \-permutation(resp.
projective for i ≤ m), there exists Qi such that Pi⊕Qi is
permutation (resp.free for i ≤ m). Adding to P ⊕ΣP short complexes
· · · 0→ Qi
1−→ Qi → 0 · · · , withQi in degrees i+ 1 and i, yields the
wanted m-free permutation complex P̃ . �
2.13. Remark. The thick closure of the subcategory P(G;R) of
Db(RG)
P(G;R)\ = Q(G;R)\ = thick(P(G;R)) = thick(Q(G;R))
is a central object in this paper, as it will turn out (Theorem
3.21) to be the essentialimage of F̄ in (1.2). We also want to
decide when P(G;R)\ coincides with Db(RG).More generally, we want
to construct an invariant on Db(RG) that detects P(G;R)
\.We sow this invariant in Section 5 and reap the result in
Theorem 6.16.
2.14. Example. For G = 1, this gives P(1;R)\ = Dperf(R), the
perfect complexes.
Let us start with generalities about the Mackey 2-functor
P(?;R).
2.15. Recollection. Recall that for a subgroup H ≤ G, induction
is a two-sided ad-joint to restriction. In particular, these
functors preserve injectives and projectivesand yield well-defined
functors on derived categories without need to derive themon either
side. Recall also that the composite (for Y a module or a complex
over H)
(2.16) Yη`//
id
OOResGH Ind
GH(Y )
�r // Y
of the (usual) unit for the Ind a Res adjunction and the (usual)
counit for theRes a Ind adjunction is the identity. The other
composite (now for X over G)
(2.17) Xηr//
[G:H]·OOInd
GH Res
GH(X)
�` // X
of the (usual) Res a Ind unit and Ind a Res counit is
multiplication by [G : H].
2.18. Proposition. Let H ≤ G be a subgroup.(a) For X ∈ Db(RG),
if X ∈ P(G;R)\ then ResGH(X) ∈ P(H;R)\.(b) For Y ∈ Db(RH), we have
Y ∈ P(H;R)\ if and only if IndGH(Y ) ∈ P(G;R)\.(c) If X ∈ Db(RG) is
such that ResGH(X) ∈ P(H;R)\ and multiplication by [G : H]
is invertible on X (e.g. if [G : H] is invertible in R) then X ∈
P(G;R)\.
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8 PAUL BALMER AND MARTIN GALLAUER
Proof. Direct from ResGH and IndGH being exact and preserving
permutation mod-
ules and free modules, using (2.16) in the ‘if’ part of (b) and
using (2.17) in (c). �
2.19. Definition. We say that a complex X ∈ Db(RG) is R-perfect
if the underlyingcomplex ResG1 (X) is perfect over R. This defines
a thick subcategory of Db(RG)
DR-perf(RG) :={X ∈ Db(RG)
∣∣ ResG1 (X) ∈ Dperf(R)}.2.20. Corollary. We have P(G;R)\ ⊆
DR-perf(RG). If the order |G| is invertiblein R, then P(G;R)\ =
DR-perf(RG).
Proof. Immediate from Proposition 2.18 for H = 1 and from
Example 2.14. �
Let us say a few words of the tensor product and its
preservation of P(G;R).For the end of the section, we allow
arbitrary modules and unbounded complexes.
2.21. Recollection. We can ‘tensor over R and use diagonal
G-action’ as usual:
−⊗R − : Mod(RG)×Mod(RG)−→Mod(RG⊗R RG)−→Mod(RG).This right-exact
tensor can be left-derived (we only use the following
generality):
−⊗LR − : D+(RG)×D+(RG)−→D+(RG).Given right-bounded complexes X ∈
Ch+(RG) and Y ∈ Ch+(RG), choose projec-tive resolutions PX → X and
PY → Y and define in D+(RG):
X ⊗LR Y := PX ⊗R PY ∼= X ⊗R PY ∼= PX ⊗R Y.The latter two
isomorphisms use that PX and PY are degreewise R-flat. In fact,
ifeither X or Y ∈ Ch+(RG) is degreewise R-flat, we have X ⊗LR Y ∼=
X ⊗R Y .
For a subgroup H ≤ G, the properties of restriction
(Recollection 2.15) give us
ResGH(X ⊗LR Y ) ∼= ResGH(X)⊗LR Res
GH(Y ).
2.22. Proposition. Let P,Q ∈ Chb(perm(G;R)). If P is m-free and
Q in n-freethen P ⊗R Q remains in Chb(perm(G;R)) and is (m+ n+
1)-free.
Proof. The tensor of permutation modules is permutation by
Mackey and the tensorof a permutation with a free is free by
Frobenius. Hence if ` ≤ m+n+1 and ` = i+jthen i ≤ m or j ≤ n and
the summand Pi ⊗R Qj of (P ⊗R Q)` is free. �
2.23. Corollary. The tensor product ⊗LR on D+(RG) restricts to a
functor
−⊗LR − : P(G;R)× P(G;R)−→P(G;R).And similarly P(G;R)\ ⊗LR
P(G;R)\ ⊆ P(G;R)\.
Proof. If s : P → X and t : Q → Y are m-free permutation
resolutions for m ≥ 0then so is s⊗LR t : P ⊗R Q ∼= P ⊗LR Q
s⊗Lt−−−→ X ⊗LR Y by Proposition 2.22. �
Here is another use of the tensor, perhaps even more
straightforward:
2.24. Proposition. Let R → R′ be a ring homomorphism. Then the
extension-of-scalars functor R′ ⊗LR − : D+(RG) → D+(R′G) maps
R-perfect complexes toR′-perfect complexes (Definition 2.19) and
sends P(G;R)\ into P(G;R′)\.
Proof. We have ResG1 (R′ ⊗LR X) ∼= R′ ⊗LR Res
GH(X) and R
′ ⊗LR − : D(R) → D(R′)preserves perfect complexes. And if P → X
is an m-free permutation resolution(over RG), then so is R′ ⊗R P ∼=
R′ ⊗LR P → R′ ⊗LR X (over R′G). �
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PERMUTATION RESOLUTIONS AND COHOMOLOGICAL SINGULARITY 9
2.25. Remark. One can also tensor over Z and use diagonal
G-action to get
−⊗Z − : Mod(ZG)×Mod(RG)−→Mod(RG)and its derived version −⊗LZ− :
D+(ZG)×D+(RG)→ D+(RG). This generalizedtensor is composed of the
tensors of Propositions 2.23 for R and 2.24 for Z→ R:
X ⊗LZ Y ∼= (R⊗LZ X)⊗LR Y.It follows in particular that we still
have P(G; Z)\ ⊗LZ P(G;R)\ ⊆ P(G;R)\.
3. Essential image of the functor F̄
Recall the thick subcategory of Db(RG) from the previous section
(Remark 2.13):
(3.1) P(G;R)\ =
{X ∈ Db(RG)
∣∣∣∣ X ⊕ ΣX admits m-free permutationresolutions (Def. 2.1) for
all m ≥ 0}.
In this section, we prove that the canonical functor F̄ of (1.2)
in the Introductionis fully-faithful, with P(G;R)\ as essential
image. See Theorem 3.21. We also provethat F̄ is essentially
surjective for R a field (Corollary 3.11) or R = Z (Lemma
3.12).This fact holds for all regular rings: see Corollary 6.20 or
Scholium 3.15.
First, we state an abstract form of Carlson [Car00], to reduce
several problems toelementary abelian groups, without repeating
ourselves. (See Remark 2.25 for ⊗LZ.)
3.2. Lemma. Suppose given for every subgroup H ≤ G two thick
subcategoriesC1(H) and C2(H) of Db(RH) satisfying the following two
conditions:
(1) The family C1(−) is closed under restriction and C2(−) under
induction.(2) For every elementary abelian subgroup E ≤ G and
Z-free ZE-modules W , we
have W ⊗LZ C1(E) ⊆ C2(E), meaning W ⊗LZ X ∈ C2(E) for all X ∈
C1(E).Then we have C1(G) ⊆ C2(G).
Proof. Let us use the same notation as [Car00, Theorem 2.1].
There exist a Z-freeZG-module V , a filtration by ZG-submodules
(for Z with trivial G-action)
0 = L0 ⊆ L1 ⊆ · · · ⊆ Lτ = Z⊕ V,elementary abelian Ei ≤ G and
Z-free ZEi-modules Wi, for 1 ≤ i ≤ τ , such thatLi/Li−1 ' IndGEiWi.
Given now X ∈ Chb(RG), define in Chb(RG) the objects
X ′ = V ⊗Z X and Ni = Li ⊗Z X for all 0 ≤ i ≤ τ.Then, by
flatness over Z of the modules Li and V , we get a filtration
0 = N0 ⊆ N1 ⊆ · · · ⊆ Nτ−1 ⊆ Nτ = X ⊕X ′
by subcomplexes such that (using Z-flatness of Wi in the third
isomorphism)
Ni/Ni−1 ' (IndGEiWi)⊗ZX ∼= IndGEi
(Wi⊗
ZResGEi(X)
) ∼= IndGEi (Wi L⊗Z ResGEi(X)).If X ∈ C1(G), it follows from (1)
and (2) that each Ni/Ni−1 belongs to C2(G) andtherefore so does X
since C2(G) is triangulated and thick in Db(RG). �
3.3. Corollary. Fix the ring R and consider the following
property of G:
(∗G) P(G;R)\ = Db(RG).If (∗E) holds for all elementary abelian
subgroups E of G, then (∗G) holds for G.
Proof. Use C1(H) = Db(RH), C2(H) = P(H;R)\ and Proposition 2.18.
�
-
10 PAUL BALMER AND MARTIN GALLAUER
We now study the effect of inverting the order of |G| (cf.
Proposition 2.18 (c)).
3.4. Lemma. Let r ∈ R and set R′ = R[1/r]. The canonical functor
R′ ⊗R − :Chb(perm(G;R))→ Chb(perm(G;R′)) is essentially
surjective.
Proof. Pick P ′ ∈ Chb(perm(G;R′)). We can assume Pi = 0 unless 0
≤ i ≤ n. EachP ′i = R
′(Ai) for some finite G-set Ai clearly comes from R(Ai) over R.
Let N ≥ 1and consider the following construction (note the changing
powers of r, vertically):
P (N) :=
s(N)��
· · · 0 // P ′n∂n //
rnN
��
P ′n−1∂n−1
//
r(n−1)N��
· · · ∂2 // P ′1∂1 //
rN
��
P ′0 //
1��
0 · · ·
P ′ = · · · 0 // P ′n∂′n
// P ′n−1∂′n−1
// · · ·∂′2
// P ′1∂′1
// P ′0 // 0 · · ·
where ∂i := rN∂′i. This is an isomorphism s(N) : P (N)
∼→ P ′ in Chb(perm(G;R′)).Increasing N � 1, one easily arranges
that the maps ∂i in P (N) are defined over R,that they are
RG-linear and finally that they form a complex, for all these
propertiesonly involve finitely many denominators. Then P (N)
provides a source of P ′. �
3.5. Proposition. Let r ∈ R and set R′ = R[1/r]. Let X ∈ Db(RG)
such thatR′ ⊗LR X ∈ P(G;R′). Then there exists an exact triangle P
→ X → T → ΣP inDb(RG) where P ∈ Chb(perm(G;R)) and T is r-torsion,
i.e. rn· idT = 0 for n� 1.
Proof. By Lemma 3.4, there exists P ∈ Chb(perm(G;R)) and an
isomorphismR′ ⊗R P
∼→ R′ ⊗R X in Db(R′G). By the usual localization sequence
Dr-torsb (RG) //incl // Db(RG)
R′⊗LR− // // Db(R′G)
(see Keller [Kel99, Lemma 1.15]), we have Db(R′G) = Db(RG)[1/r].
So the isomor-
phism R′⊗RP∼→ R′⊗RX is given by a fraction P
rn←− P f−→ X in Db(RG) for somen� 1 and some f : P → X such that
T := cone(f) belongs to Dr-torsb (RG). �
3.6. Corollary. Let G be a p-group and X ∈ Db(RG) be R-perfect
(Definition 2.19).Then there exists an exact triangle in Db(RG)
P → X ⊕ ΣX → T → ΣPwhere P belongs to Chb(perm(G;R)) and T is
p-torsion.
Proof. We apply Proposition 3.5 for r = p, so R′ = R[1/p]. It is
easy to check thatX ′ := R′ ⊗R X ∈ Db(R′G) remains R′-perfect. By
Corollary 2.20, we have X ∈P(G;R′)\ hence X⊕ΣX ∈ P(G;R′). The
result then follows by Proposition 3.5. �
3.7. Remark. If X is p-torsion, say pn · idX = 0, then the
octahedron axiom gives
X ∈ thick(X ⊕ ΣX) = thick(cone(X pn
−→ X)) ⊆ thick(cone(X p−→ X)).
Adapting [BB20] let us now see when the trivialRG-moduleR
belongs to P(G;R).
3.8. Proposition. The following are equivalent:
(i) The trivial RG-module R belongs to P(G;R).(ii) It admits a
0-free permutation resolution (Lemma 2.4): There is a
resolution
0→ Pn → · · · → P1 → P0 → R→ 0 by permutation RG-modules with P0
free.
Proof. (i)⇒(ii) is trivial. (ii)⇒(i) by Proposition 2.22 and R⊗m
∼= R, ∀m ≥ 0. �
-
PERMUTATION RESOLUTIONS AND COHOMOLOGICAL SINGULARITY 11
3.9. Proposition. Let G be an abelian group. Then R belongs to
P(G;R).
Proof. Consider first the cyclic group Cn = 〈x | xn = 1 〉 of
order n, so thatRG = R[x]/(xn − 1). Then the sequence of
RCn-modules
0 // R(xn−1+···+x+1)·
// R[x]/(xn − 1)(x−1)·
// R[x]/(xn − 1) x 7→1 // R // 0
is easily verified to be exact, with both modules R acted upon
trivially.In general, since G is a product of cyclic groups, we
find subgroups H1, . . . ,Hr
of G such that each G/Hi is cyclic and H1 ∩ · · · ∩Hr = 1.
Inflating from the cyclicgroups G/Hi the above resolution, we get
quasi-isomorphisms
Q(i) :=si ��
0 // R //
��
R(G/Hi) //
��
R(G/Hi) //
��
0
1 = 0 // 0 // 0 // R // 0
for i = 1, . . . , r. Then s1 ⊗ · · · ⊗ sr : Q(1) ⊗R · · · ⊗R
Q(r) → 1 gives a 0-freepermutation resolution by the Mackey
formula. Then apply Proposition 3.8. �
3.10. Lemma. Let G be an abelian group. Then every bounded
complex of permu-tation RG-modules belongs to P(G;R).
Proof. Let H ≤ G be a subgroup. By Propositions 3.9 and 2.18, we
have R(G/H) =IndGH(R) ∈ Ind
GH(P(H;R)) ⊆ P(G;R). Finally, P(G;R) is triangulated. �
3.11. Corollary. Let k be a field. Then P(G; k)\ = Q(G; k)\ =
Db(kG).
Proof. By Corollary 3.3, we can assume that G is an elementary
abelian p-group. Ifp is invertible in k, we are done by Corollary
2.20. If char(k) = p then k ∈ P(G; k)since G is abelian
(Proposition 3.9) and thick(k) = Db(kG) since G is a p-group. �
3.12. Lemma. We have P(G; Z)\ = Db(ZG).
Proof. As before, we reduce to the case of G an elementary
abelian p-group byCorollary 3.3, for some prime p. Pick X ∈ Db(ZG).
Note that X is Z-perfect and,by Corollary 3.6, there exists an
exact triangle
P → X ⊕ ΣX → T → ΣPin Db(ZG) where P is a complex of permutation
modules, and T is p-torsion. Inparticular P ∈ P(G; Z) already, by
Lemma 3.10. So it suffices to prove T ∈ P(G; Z)\,i.e. we can assume
that pn ·X = 0 for some n � 1. By Remark 3.7 we then haveX ∈
thick(cone(X p−→ X)). We compute cone(X p−→ X) ∼= cone(Z p−→ Z) ⊗LZ
X ∼=i∗Li
∗(X) by the projection formula for the adjunction
Db(ZG)
Li∗=Fp⊗LZ− ��Db(FpG)
i∗
OO
given by the usual extension and restriction of scalars along Z→
Fp = Z/p. Hence(3.13) X ∈ i∗(Db(FpG)).Using that thick(Fp) =
Db(FpG) since G is a p-group we continue:
(3.14) X ∈ i∗(thick(Fp)) ⊆ thick(i∗(Fp)) = thick(cone(Zp−→ Z)) ⊆
thick(Z).
And we conclude from Z ∈ P(G; Z), as already established in
Proposition 3.9. �
-
12 PAUL BALMER AND MARTIN GALLAUER
3.15. Scholium. The above proof almost works for any regular
ring R instead of Z.Of course Fp needs to be replaced by R̄ = R/p.
Using regularity, we can go upto (3.13) in the above proof. One
point which is more complicated is that R̄ doesnot necessarily
generate Db(R̄G), as we used to pass to (3.14) for Fp. (Here R̄
isnot necessarily regular.) Instead, for G a p-group, one can show
that the image of
Db(R̄) under InflG1 (i.e. R̄-modules with trivial G-action)
generates Db(R̄G). This
uses nilpotence of the augmentation ideal I := Ker(R̄G → R̄),
and the associ-ated finite filtration · · · I`+1N ⊆ I`N · · · of
any R̄G-module N , in which everyI`N/I`+1N has trivial G-action.
Then one uses regularity of R one more time andthe commutativity of
the following square
(3.16)
Db(R̄)i∗ //
InflG1��
Db(R) = Dperf(R) = thick(R)
InflG1��
Db(R̄G)i∗ // Db(RG)
to conclude via Proposition 3.9 again but with the following
replacement of (3.14):
X ∈ thick(i∗(InflG1 Db(R̄))) ⊆(3.16)
thick(InflG1 R) ⊆Prop.3.9
P(G;R)\.
Returning to arbitrary noetherian commutative rings R, we extend
Lemma 3.10.
3.17. Corollary. Each bounded complex of permutation RG-modules
is in P(G;R)\.
Proof. Let H ≤ G be a subgroup. By Lemma 3.12, we know that Z ∈
P(H; Z)\. ByProposition 2.24 for the unique homomorphism Z→ R, we
get that R ∈ P(H;R)\.By Proposition 2.18, we deduce that R(G/H) =
IndGH(R) ∈ P(G;R)\. We thenconclude since P(G;R)\ ⊆ Db(RG) is thick
and triangulated (Proposition 2.9). �
We also use later the following technical consequence (see
Remark 2.25 for ⊗LZ):
3.18. Lemma. Let W ∈ Db(ZG) and X ∈ P(G;R)\. Then W ⊗LZ X ∈
P(G;R)\.
Proof. We have Db(Z)⊗LZ P(G;R)\ =3.12P(G; Z)\ ⊗LZ P(G;R)\ ⊆
2.25P(G;R)\. �
3.19. Corollary. Let X be an object of Db(RG). The following are
equivalent:
(i) The complex X belongs to P(G;R)\.
(ii) For every Sylow subgroup H ≤ G, the restriction ResGH(X)
belongs to P(H;R)\.(iii) For every elementary abelian subgroup E ≤
G, we have ResGE(X) ∈ P(E;R)\.
Proof. (i)⇒(ii)⇒(iii) are immediate from stability under
restriction as in Proposi-tion 2.18 (a). To prove (iii)⇒(i), we
apply Lemma 3.2 to C2(H) = P(H;R)\ andC1(H) =
{X ∈ Db(RH)
∣∣ ResHE (X) ∈ P(E;R)\ for all elementary abelian E ≤ H }for H ≤
G. Condition (1) is satisfied by Proposition 2.18. To verify
Condition (2),let E ≤ G be elementary abelian. In that case C1(E) =
C2(E) and Condition (2)holds by Lemma 3.18. Hence Lemma 3.2 tells
us C1(G) ⊆ C2(G). �
3.20. Remark. In Corollary 3.19, once we know that ResG1 (X) is
perfect, it suffices
to check (ii) or (iii) for the p-subgroups H or E for those
primes p such that Xp−→ X
is not invertible (in particular not invertible in R). This
holds by Corollary 2.20.
We now give a more conceptual description of the subcategory
P(G;R)\ of (3.1):
-
PERMUTATION RESOLUTIONS AND COHOMOLOGICAL SINGULARITY 13
3.21. Theorem. Let G be a finite group and R a commutative
noetherian ring.The canonical functor F̄ of (1.2) restricts to an
exact equivalence
(3.22) F̄ :
(Kb(perm(G;R))
Kb,ac(perm(G;R))
)\'−−→ P(G;R)\.
In particular, P(G;R)\ is the essential image of F̄ in
Db(RG).
Proof. By Corollary 3.17, the canonical functor Kb(perm(G;R))→
Db(mod(RG))lands inside P(G;R)\. Since the latter is
idempotent-complete there exists a well-defined exact functor F̄ as
in (3.22). By Definition 2.1 (for m = 0), it is clear thatF̄ is
surjective-up-to-direct-summands, hence it suffices to prove that
the functor
K̄ :=Kb(perm(G;R))
Kb,ac(perm(G;R))
F̄−−→ Db(RG)
is fully faithful. As every X is a retract of X ⊕ ΣX, it
suffices to prove that thehomomorphism F̄ : HomK̄(X ⊕ ΣX,Y ) →
HomDb(RG)(X ⊕ ΣX,Y ) is an isomor-phism for every X,Y ∈
Kb(perm(G;R)). By Corollary 3.17, we know that suchcomplex X
belongs to P(G;R)\ hence X ⊕ ΣX ∈ P(G;R). So it suffices to
showthat for every X,Y ∈ Kb(perm(G;R)) such that X ∈ P(G;R), the
homomorphism
F̄ : HomK̄(X,Y )→ HomDb(RG)(X,Y )
is an isomorphism. For surjectivity, let fs−1 : X → Y be
represented by a frac-tion X
s←− Z f−→ Y in Chb(RG) where s is a quasi-isomorphism. Since X
belongsto P(G;R), so does Z by Proposition 2.7. So Z admits a
0-free permutation res-olution, i.e. there exists a
quasi-isomorphism t : P → Z with P ∈ Chb(perm(G;R)).Hence our
morphism f s−1 = (ft)(st)−1 comes fromX
st←− P ft−→ Y in HomK̄(X,Y ).Injectivity is similar (or follows
from conservativity and fullness of F̄ ). �
4. Big singularity category
The target of our ‘cohomological singularity’ functor χG :
D(RG)→ Dsing(R) isthe big singularity category of the coefficient
ring R. Let us remind the reader.
4.1. Recollection. As in [Kra05], let A be a locally noetherian
Grothendieck categorycompactly generated derived category. For A =
Mod(R), the noetherian objectsnoethA is mod(R) and D(A) is
generated by D(A)c = Dperf(R). Similarly for A =Mod(RG). The big
singularity category (or stable derived category) of A is
Dsing(A) = Kac(InjA),
the full subcategory of the big homotopy category of injectives
K(InjA) spannedby acyclic complexes. There is a recollement Qλ a Q
a Qρ and Iλ a I a Iρ
(4.2)
D(A)��
Q�
��Qρ��
K(InjA)
Q
OOOO
I���
I��
Dsing(A) = Kac(InjA)
OOI
OO
for I : Kac(InjA)�K(InjA) the inclusion and Q : K(InjA)�D(A) the
localization.
-
14 PAUL BALMER AND MARTIN GALLAUER
The singularity functor (Krause’s stabilization functor) is
defined as
singA : D(A)Iλ◦Qρ−−−−→ Dsing(A).
There is a natural transformation Qλ → Qρ that is invertible on
compacts:(4.3) Qλ(X) ∼= Qρ(X) if X ∈ D(A)c
by [Kra05, Lemma 5.2]. On the larger subcategory Db(noethA), the
right adjointQρ defines an inverse to the equivalence of [Kra05, §
2] identifying K(InjA)c
Q : K(InjA)c∼→ Db(noethA).
In summary, we have a finite localization sequence D(A)Qλ�
K(InjA)
Iλ� Dsing(A)
as in (4.2) and the triangulated category Dsing(A) is compactly
generated withcompact part the (usual) small singularity category,
idempotent-completed
Dsing(A)c ∼=(
K(InjA)c
(Qλ D(A))c
)\∼=(
Db(noethA)
D(A)c
)\= Dsingb (A)
\.
Finally, we note that since K(InjA) is compactly generated and
since the inclusionK(InjA)�K(A), that we denote J , preserves
products and coproducts (because Ais locally noetherian), there is
another useful triple of adjoints Jλ a J a Jρ:
(4.4)
K(A)
J���
J��
K(InjA).
OOJ
OO
4.5. Lemma. Keep the above notation, e.g. for A = Mod(RG). Let X
be in K(A).
(a) The object Qρ(X) ∈ K(InjA) in (4.2) is a K-injective
resolution of X. Inparticular, if X ∈ K−(A) is left-bounded then
Qρ(X) belongs to K−(InjA).
(b) If X ∈ K−(A) is left-bounded, the object Jλ(X) ∈ K(InjA) in
(4.4) is an in-jective resolution of X. Hence if X ∈ K−,ac(A) is
also acyclic then Jλ(X) = 0.
(c) The restrictions of the two functors Qρ and Jλ to K−(A) are
isomorphic.
Proof. By [Kra05, Remark 3.7], we have QJλ ∼= Q+, where Q+ :
K(A)�D(A) isthe (Bousfield) localization defining D(A). It follows
that J Qρ is right adjointto Q+. So, if we let i := JQρ ◦Q+, every
X ∈ K(A) fits in an exact triangle
(4.6) a(X)→ X η−→ i(X)→ Σa(X)in K(A), where a(X) ∈ Ker(Q+) =
Kac(A) and i(X) belongs to Ker(Q+)⊥ =Kac(A)
⊥ =: KInj(A), that is, i(X) is K-injective by definition. In
other words,(4.6) is the essentially unique triangle providing the
K-injective resolution of X(see [Kra05, Corollary 3.9] if
necessary). Suppressing the functors J and Q+ thatare just the
identity on objects, we have i(X) = Qρ(X), which gives (a).
Let now A ∈ K−,ac(A). The unit η′ : A→ JJλ(A) is a map from a
left-boundedacyclic to a complex of injectives, hence η′ = 0 in
K(A). But Jλ(η
′) : Jλ∼→ JλJJλ
is invertible (J being fully faithful). Thus Jλ(A) = 0, as in
the second claim of (b).Take now X ∈ K−(A) arbitrary and an
injective resolution i(X) ∈ K−(InjA).
There is a triangle (4.6) with a(X) exact and left-bounded since
X and i(X) are. Bythe above for A = a(X), we already know that
Jλ(a(X)) = 0. Applying Jλ to thetriangle (4.6) we get Jλ(X) ∼=
Jλ(i(X)) ∼= i(X) since i(X) ∈ K(InjA). Hence (b).
Part (c) is now immediate from the uniqueness of K-injective
resolutions. �
-
PERMUTATION RESOLUTIONS AND COHOMOLOGICAL SINGULARITY 15
Let us now specialize to A = Mod(R).
4.7. Lemma. Let X be an object of D(R). Then sing(X) = 0 in
Dsing(R) if andonly if Qρ(X) belongs to the localizing subcategory
of K(Inj(R)) generated by Qρ(R).
Proof. We have sing = Iλ ◦Qρ by definition. So we have sing(X) =
0 if and only ifQρ(X) ∈ Ker(Iλ) and by (4.2) that kernel is Ker(Iλ)
= Qλ(D(R)) = Qλ(Loc(R)) =Loc(Qλ(R)), where the last equality holds
since Qλ is coproduct-preserving andfully-faithful. We conclude
from Qλ(R) ∼= Qρ(R) since R ∈ D(R)c; see (4.3). �
4.8. Remark. In fact, sing(X) = 0 is also equivalent to Qλ(X)∼→
Qρ(X).
We now describe a class of not necessarily perfect complexes in
Ker(sing). AsQρ and sing = Iλ ◦Qρ may not commute with coproducts,
we need to be careful.
4.9. Remark. We say that a family of complexes {X(i)}i∈I in
Ch(R) is degreewisefinite if for every n ∈ Z, the set
{i ∈ I
∣∣X(i)n 6= 0} is finite. Since (co)productsin D(R) are computed
degreewise, the canonical map
∐i∈I X(i) →
∏i∈I X(i) is
an isomorphism in that case and only then do we write ⊕i∈IX(i)
to refer to bothcoproduct and product. If all X(i) are complexes of
injectives, then so is ⊕i∈IX(i).
4.10. Definition. An object X ∈ D(R) is weakly perfect if there
exists an isomor-phism X ' ⊕i∈NP (i) in D(R) for a countable family
{P (i)}i∈N of strictly perfectcomplexes P (i) ∈ Chb(proj(R)) such
that P (i)n = 0 unless ai ≥ n ≥ bi and suchthat limi→∞ ai = −∞.
(Such a family {P (i)}i∈N is necessarily degreewise finite.)
4.11. Example. For any P perfect, the complex X =∐i≥0 Σ
−iP is weakly perfect.
4.12. Lemma. Let X ∈ D(R) be weakly perfect. Then sing(X) = 0 in
Dsing(R).
Proof. Let X ' ⊕i∈NP (i) for {P (i)}i∈N as in Definition 4.10.
By Lemma 4.5 (a),P (i) → Qρ(P (i)) are injective resolutions and
thus live in degrees ≤ ai. Sothe family {Qρ(P (i))}i∈N is
degreewise finite (Remark 4.9) and
∐i∈N Qρ(P (i))
∼=∏i∈N Qρ(P (i)) in K(Inj(R)). Since the right adjoint Qρ
preserves products, we get
Qρ(X) ' Qρ(∏i∈N
P (i)) ∼=∏i∈N
Qρ(P (i)) ∼=∐i∈N
Qρ(P (i)).
Hence Qρ(X) belongs to Loc(Qρ(Dperf(R))) and sing(X) = 0 by
Lemma 4.7. �
4.13. Lemma. Let X,Y ∈ D(R) be weakly perfect. Then X⊗LR Y is
weakly perfect.
Proof. Say X ' ⊕i∈NP (i) and Y ' ⊕j∈NQ(j) with P (i) and Q(j)
strictly perfectcomplexes, respectively concentrated in degrees
between ai and bi and between cjand dj , and such that limi ai =
limj cj = −∞. Then X ⊗LR Y ' qi,jP (i)⊗R Q(j),each P (i)⊗RQ(j) is
strictly perfect and concentrated in degrees between ai+cj andbi+dj
. As N×N is countable and for each n ∈ Z the set
{(i, j) ∈ N×N
∣∣ ai+cj ≥ n}is finite (otherwise limi ai = limj cj = −∞ would
fail), we get the result. �
5. Cohomological singularity
In this section, we define the announced cohomological
singularity functor (1.5).
5.1. Recollection. The functor that equips every R-module with
trivial G-action
InflG1∼= homR(R,−) ∼= R⊗R − : Mod(R)→ Mod(RG).
-
16 PAUL BALMER AND MARTIN GALLAUER
has adjoints the usual G-orbits (−)G and G-fixed points (−)G
(5.2)
Mod(RG)
R⊗RG−= (−)G""
homRG(R,−) = (−)G||
Mod(R)
InflG1
OO
This triple of adjoints passes to homotopy categories of
complexes on the nose. Forderived categories, we left-derive the
left adjoint and right-derive the right one:
(5.3)
D(RG)
R⊗LRG−=:(−)hG""
RhomRG(R,−)=:(−)hG||
D(R)
InflG1
OO
So (−)hG provides a complex whose homology groups are
G-cohomology as in (1.6).
5.4. Definition. Let H ≤ G be a subgroup. The H-cohomological
singularity func-tor χH = singR ◦(−)hH is the following composite
(see Recollection 4.1 for sing):
χH : D(RG)ResGH−−−→ D(RH) (−)
hH
−−−−→ D(R) singR−−−→ Dsing(R).
We say that a complex X ∈ D(RG) is H-cohomologically perfect if
χH(X) = 0.We say that X is cohomologically perfect, if it is
H-cohomologically perfect for allsubgroups H ≤ G, that is, if ⊕H
χH(X) = 0 in Dsing(R).
5.5. Remark. We remind the reader that although Ker(sing)∩Db(RG)
= Dperf(R),the kernel of sing : D(R) → Dsing(R) on the big derived
category is larger thanDperf(R); see for instance Lemma 4.12. So
even whenH = 1, being 1-cohomologicallyperfect is more flexible
than being R-perfect (Definition 2.19) although the two no-
tions coincide when X ∈ Db(RG) is bounded, i.e. when ResG1 (X) ∈
Db(R).For more general subgroups H ≤ G, even a bounded complex X ∈
Db(RG) can
be H-cohomologically perfect without XhH being perfect; see
Example 5.9.We provide a better justification of the terminology in
Remark 6.19.
5.6. Remark. The functor (−)G ∼= homRG(R,−) is a special value
of the bifunctor
Mod(RG)op ×Mod(RG) homRG(−,−)−−−−−−−−→ Mod(R).
It follows that for any X ∈ D(RG) the objet XhG is represented
by both
(5.7) homRG(PR, X) and homRG(R, i(X))
where PR → R is a projective resolution of R as an RG-module,
and X → i(X) isa K-injective resolution of X, for both are
quasi-isomorphic to homRG(PR, i(X)).
5.8. Remark. If the order |G| is invertible in R, then the
trivial RG-module R isprojective by (2.17). In that case, (−)G is
exact and coincides with (−)hG.
We can use this to see that being G-cohomologically perfect does
not imply beingH-cohomologically perfect for each subgroup H ≤ G,
even for H = 1.
5.9. Example. Let R = Z/9 and G = C2 = 〈x | x2 = 1 〉. Consider
the R-module M = Z/3 with the action of x by −1. We have MG = 0
hence MhG = 0by Remark 5.8. In particular, M is G-cohomologically
perfect but it is not H-cohomologically perfect for the subgroup H
= 1 since Z/3 is not perfect over Z/9.
-
PERMUTATION RESOLUTIONS AND COHOMOLOGICAL SINGULARITY 17
Let us establish some generalities about the cohomological
singularity functor.
5.10. Proposition. Let H ≤ G be a subgroup. There are canonical
isomorphisms(−)hG ◦ IndGH ∼= (−)hH and χG ◦ Ind
GH∼= χH .
Proof. This first follows from the relation ResGH ◦ InflG1 =
Infl
H1 , by taking right
adjoints and right-deriving. The second follows by
post-composing with singR. �
5.11. Corollary. Let H ≤ G. Then induction IndGH : Db(RH) →
Db(RG) andrestriction ResGH : Db(RG)→ Db(RH) preserve
cohomologically perfect complexes.
Proof. Restriction is built into Definition 5.4. For induction,
it follows immediatelyfrom the Mackey formula and Proposition 5.10.
�
5.12. Lemma. Let G be an elementary abelian group and X ∈
P(G;R)\; see (3.1).Then X is cohomologically perfect.
Proof. By Theorem 3.21, it suffices to prove that R(G/H) is
cohomologically perfectfor every H ≤ G. By Mackey and Proposition
5.10, it suffices to show χG(R) = 0.
Assume first that G = Cp = 〈x | xp = 1 〉 is cyclic of order p.
We compute RhGusing (5.7). Let PR be the usual projective
resolution of R:
(5.13) · · · → RCpx−1−−−→ RCp
xp−1+···+1−−−−−−−→ RCpx−1−−−→ RCp → 0
Applying homRCp(−, R), we conclude that RhG is represented by
the complex
0→ R 0−→ R p−→ R 0−→ R p−→ R 0−→ R→ · · · ,which is weakly
perfect in the sense of Definition 4.10.
In general, let G = C×rp . Then RG = (RCp) ⊗R · · · ⊗R (RCp),
with r factors.Let P
(i)R be the inflation of (5.13) along the projection G→ Cp onto
the ith factor.
One checks that one has a canonical isomorphism of complexes of
R-modules
homRG(P(1)R , R)⊗R · · · ⊗R homRG(P
(r)R , R)
∼→ homRG(P (1)R ⊗R · · · ⊗R P(r)R , R).
The tensor product ⊗ri=1P(i)R on the right-hand side is a free
resolution of R as
an RG-module hence the right-hand side computes RhG. As
homRG(P(i)R , R)
∼=homRCp(PR, R) is weakly perfect over R by the above special
case, we conclude via
Lemma 4.13 that RhG is weakly perfect. Thus sing(RhG) = 0 by
Lemma 4.12. �
We can now extend Lemma 5.12 to any finite group G.
5.14. Proposition. Every object of P(G;R)\ is cohomologically
perfect.
Proof. We apply Lemma 3.2 with C1(G) = P(G;R)\ and C2(G) the
thick sub-
category of Db(RG) spanned by cohomologically perfect complexes.
The formeris closed under restriction by Proposition 2.18, and the
latter is closed under in-duction by Corollary 5.11. This verifies
Condition (1) of Lemma 3.2. For Con-dition (2), let E ≤ G be
elementary abelian and W a Z-free ZE-module andX ∈ C1(E) = P(E;R)\.
By Lemma 3.18, we have W ⊗LZ X ∈ P(E;R)\ ⊆ C2(E)where the latter
inclusion holds by Lemma 5.12. Hence Condition (2) holds too. �
5.15. Corollary. Every bounded complex X of permutation
RG-modules is coho-mologically perfect.
Proof. By Corollary 3.17 we have X ∈ P(G;R)\ and then use
Proposition 5.14. �
-
18 PAUL BALMER AND MARTIN GALLAUER
We can now apply Proposition 5.14 to show that beingR-perfect
(Definition 2.19)is not sufficient to belong to Im(F̄ ) = P(G;R)\.
Compare Corollary 2.20.
5.16. Example. Let k = F2 and consider the ring R = k[x]/〈x2 −
1〉. Take G =C2 = 〈 y | y2 = 1 〉 cyclic of order 2. Let X = Rx
denote the ring R viewed as anRG-module with the non-trivial action
of y via x. This X ∈ Db(RC2) is R-perfectbut we claim that χC2(X)
6= 0. As RC2 is self-injective, the following resolution
0→ Rxy−x−−−→ RC2
y−x−−−→ RC2 → · · ·is an injective resolution of Rx. Computing
(Rx)
hG in D(R) as in (5.7) with the
above i(X) = · · · 0→ RC2y−x−−−→ RC2
y−x−−−→ RC2 → · · · , we get that (Rx)hG is
· · · 0→ R 1−x−−−→ R 1−x−−−→ R→ · · ·and we deduce that (Rx)
hG ' k in D(R). But k ∈ Db(R) is not perfect, henceχG(X) '
sing(k) 6= 0. Using Proposition 5.14, this means X /∈ P(G;R)\.
6. Main result
We have established in Theorem 3.21 that the canonical functor
(1.2)
F̄ :
(Kb(perm(G;R))
Kb,ac(perm(G;R))
)\−→Db(RG)
is fully faithful and has essential image the auxiliary category
P(G;R)\ of (3.1).We saw in Proposition 5.14 that P(G;R)\ is
contained in the subcategory of coho-mologically perfect complexes
(Definition 5.4). We now want to prove the converse.
Two ideas will be key: the “cohomology” comonad InflG1 ◦(−)hG on
cohomologi-cally perfect objects, and compactness arguments. To
make both work at the sametime, we lift that comonad to the
homotopy category of injectives, K(Inj(RG)),whose compact part is
the bounded derived category. The proof of our main resultbeing
pretty long, we prove several shorter lemmas. Let us first set the
notation.
6.1. Recollection. We are going to construct the following
diagram via [Kra05, § 6]
(6.2)
K(Inj(RG))
ĉ!
��
Qtttt
D(RG)
44
Qλ 44
44Qρ
44
c!
��
K(Inj(R))
ĉ∗
OO
Qtttt
D(R)
44
Qλ 44
44 Qρ
44
c∗
OO
We already encountered the slanted arrows Qλ a Q a Qρ in the
recollement (4.2).The left-hand vertical arrows c∗ a c! are simply
a shorthand for (5.2):
c∗ := InflG1 and c
! := (−)hG.There are several reasons for this notation. First,
it is lighter in formulas involvingiterated compositions. Second,
it evokes the algebro-geometric notation c∗ a c∗ a c!for an
imaginary closed immersion c : Spec(R) ↪→ Spec(RG) – that actually
makessense if G is abelian. (And we do have a left adjoint c∗ too,
namely the left-derivedfunctor of deflation (−)hG.) Finally, it
allows for a simple notation at the level ofK(Inj), namely the
yet-to-be-explained ĉ∗ a ĉ! on the right-hand side of (6.2).
-
PERMUTATION RESOLUTIONS AND COHOMOLOGICAL SINGULARITY 19
For this, we apply [Kra05, § 6] to the exact functor (denoted F
in loc. cit.)InflG1 : Mod(R) → Mod(RG). Its right adjoint (−)G :
Mod(RG) → Mod(R) pre-serves injectives and our ĉ! : K(Inj(RG)) →
K(Inj(R)) is simply (−)G degreewise.Its left adjoint ĉ∗ :
K(Inj(R)) → K(Inj(RG)) is more subtle than just inflation. Itis
Krause’s construction, namely ĉ∗ is defined as the following
composite
ĉ∗ : K(Inj(R))J� K(Mod(R))
InflG1−−−−→ K(Mod(RG))Jλ� K(Inj(RG)).
where J : K(Inj)�K(Mod) is the inclusion and Jλ : K(Mod)�K(Inj)
its left ad-joint, as in (4.4). It is an easy exercise to verify
(using J fully faithful) that ĉ∗ a ĉ!.(Although we had a derived
left adjoint c∗ a c∗ there is no ĉ∗ a ĉ∗ on K(Inj).)
By [Kra05, Lemma 6.3], since inflation is exact, we have
(6.3) Q ◦ ĉ∗ ∼= c∗ ◦Q : K(Inj(R))→ D(RG).
From this we deduce, by taking right adjoints, that
(6.4) ĉ! ◦Qρ ∼= Qρ ◦ c!.
Note that since the functor (−)G : Mod(RG) → Mod(R) preserves
coproducts,so does the induced ĉ! on K(Inj). Thus its left adjoint
preserves compacts:
(6.5) ĉ∗(K(Inj(R))c) ⊆ K(Inj(RG))c.
6.6. Remark. On every Y ∈ K(Inj(RG)) the comonad ĉ∗ĉ! equals
by construction
ĉ∗ĉ!(Y ) = Jλ Infl
G1 J ĉ
!(Y ) = Jλ InflG1 (J(Y )
G) ∼= Jλ homRG(R, Y )
where homRG(R, Y ) has trivial G-action. This leads us to
bimodule actions:
6.7. Lemma. There is an action of the bounded derived category
of R(G × Gop)-modules on K−(Inj(RG)), in the form of a well-defined
bi-exact functor
[−,−] : Db(R(G×Gop))op ×K−(Inj(RG))→ K−(Inj(RG))
given by the formula [L, Y ] = Jλ(homRG(L, Y )).
Proof. On the level of module categories, there is an action
homRG(−,−) : Mod(R(G×Gop))op ×Mod(RG)→ Mod(RG)
which takes (L, Y ) to the abelian group HomRG(L, Y ) built by
viewing L as a leftRG-module via its left G-action, and then making
the output HomRG(L, Y ) intoa left G-module homRG(L, Y ) by using
the ‘remaining’ right G-action on L. Beingadditive in both
variables, this passes to homotopy categories
(6.8) homRG(−,−) : K(R(G×Gop))op ×K(RG)→ K(RG)
(by totalizing via∏
, which is irrelevant in our bounded case). This yields
[−,−] : Kb(R(G×Gop))op ×K−(Inj(RG))(6.8)−−−→ K−(RG)
Jλ−→ K−(Inj(RG)).
The preservation of left-boundedness by Jλ is Lemma 4.5 (b). To
show that thisdescends to the derived category in the first
variable, let L ∈ Kb(R(G × Gop)) beacyclic and let Y ∈ K−(Inj(RG)),
and let us show that Jλ(homRG(L, Y )) = 0.By Lemma 4.5 (b) again,
it suffices to show that homRG(L, Y ) is acyclic. ButHn homRG(L, Y
) = HomK(RG)(L[n], Y ) vanishes since Y is a left-bounded complexof
injectives and L acyclic (as complex of RG-modules as well). �
-
20 PAUL BALMER AND MARTIN GALLAUER
6.9. Remark. Each object L in Db(R(G×Gop)) thus defines an exact
endofunctor[L,−] : K−(Inj(RG))→ K−(Inj(RG)).
For instance, [RG,−] ∼= Id whereas [R,−] ∼= ĉ∗ĉ! is our
comonad, by Remark 6.6.We use this to decide when Y ∈ K−(Inj(RG))
can be recovered from ĉ∗ĉ!(Y ).
6.10. Lemma. Let G be a finite p-group and Y ∈ K−(Inj(RG)),
left-bounded andsuch that pn ·idY = 0 for n� 1. Then Y belongs to
thick(ĉ∗ĉ!(Y )) in K−(Inj(RG)).
Proof. As explained in Remark 6.9, we need to show that in
K−(Inj(RG))
[RG, Y ] ∈ thick([R, Y ]).Since pn · Y = 0, we also have pn ·
[RG, Y ] = 0 and Remark 3.7 gives
[RG, Y ] ∈ thick(cone([RG, Y ] p−→ [RG, Y ])) = thick([cone(RG
p−→ RG), Y ])using exactness of [−, Y ]. Hence it suffices to prove
in Db(R(G×Gop)) that
cone(RGp−→ RG) ∈ thick(R).
By Proposition 2.24 for Z→ R, it suffices to prove that in
Db(Z(G×Gop))
cone(ZGp−→ ZG) ∈ thick(Z).
Consider the exact functor i∗ : Db(Fp(G × Gop)) → Db(Z(G ×
Gop)). The abovecone(ZG
p−→ ZG) is nothing but i∗(FpG) and i∗(Fp) ∼= cone(Zp−→ Z)
belongs
to thick(Z). So we are reduced to show that FpG ∈ thick(Fp) in
Db(Fp(G×Gop)),which is clear since Db(Fp(G×Gop)) = thick(Fp) as
G×Gop is a also p-group. �
6.11. Lemma. Let G be a p-group. Let X ∈ Db(RG) be p-torsion
(Proposition 3.5)and G-cohomologically perfect (Definition 5.4).
Then X belongs to thick(c∗R).
Proof. By Lemma 4.7, the assumption 0 = χG(X) = IλQρ c!(X)
implies that
(6.12) Qρ c!(X) ∈ Loc(Qρ(R))
in K(Inj(R)). Applying to this relation the
(coproduct-preserving) left adjointĉ∗ : K(Inj(R))→ K(Inj(RG)) of
(6.2), we obtain in K(Inj(RG))
ĉ∗ ĉ!Qρ(X) ∼=
(6.4)ĉ∗Qρ c
!(X) ∈(6.12)
Loc(ĉ∗Qρ(R)).
Hence by Lemma 6.10 with Y = Qρ(X), which is p-torsion since X
is, we have
(6.13) Qρ(X) ∈ thick(ĉ∗ĉ!Qρ(X)) ⊆ Loc(ĉ∗Qρ(R))in K(Inj(RG)).
Now Qρ(X) is compact in K(Inj(RG)) by Recollection 4.1 andĉ∗Qρ(R)
is compact because Qρ(R) is and ĉ∗ preserves compacts (6.5). So,
by[Nee92, Lemma 2.2], we can replace ‘Loc’ by ‘thick’ in (6.13),
giving us the relation
Qρ(X) ∈ thick(ĉ∗Qρ(R)
)in K(Inj(RG)). Applying Q : K(Inj(RG))�D(RG) and QQρ ∼= Id, we
get
X ∈ thick(Qĉ∗Qρ(R)
) (6.3)= thick
(c∗QQρ(R)
)= thick
(c∗(R)
). �
6.14. Lemma. Let G be a p-group and X ∈ Db(RG). The following
are equivalent:(i) X ∈ P(G;R)\; see (3.1).
(ii) X is cohomologically perfect (Definition 5.4).(iii) X is
G-cohomologically perfect and R-perfect (Definition 2.19).
-
PERMUTATION RESOLUTIONS AND COHOMOLOGICAL SINGULARITY 21
Proof. The implication (i)⇒(ii) is Proposition 5.14, and the
implication (ii)⇒(iii)is trivial by Definition 5.4. For the
implication (iii)⇒(i) suppose that χG(X) = 0and X is R-perfect. By
Corollary 3.6, there exists an exact triangle in Db(RG)
P → X ⊕ ΣX → T → ΣPwhere P is a bounded complex of permutation
modules (hence belongs to P(G;R)\
by Corollary 3.17) and where T ∈ Db(RG) is p-torsion. Note that
T remains G-cohomologically perfect and R-perfect since P and X
are. In other words, we reduceto the case where X is p-torsion. But
then Lemma 6.11 tells us that X ∈ thick(c∗R).We conclude by
Corollary 3.17 which guarantees that c∗(R) ∈ P(G;R)\. �
6.15. Remark. For G a p-group the equivalence (ii)⇔(iii) in
Lemma 6.14 shows thatG-cohomological perfection together with
R-perfection does imply H-cohomologicalperfection for all H ≤ G.
This is sharp by Example 5.9 and Example 5.16.
Here is the main result. The functor F̄ is in (1.2). The
subcategory P(G;R)\ ⊆Db(RG) is in (3.1). The invariant χ
H is in Definition 5.4. See also Theorem 3.21.
6.16. Theorem. Let G be a finite group and R a commutative
noetherian ring. LetX ∈ Db(RG) be a bounded complex. The following
properties of X are equivalent:
(i) The complex X belongs to Im(F̄ ) = P(G;R)\.
(ii) It is cohomologically perfect: χH(X) = 0 in Dsing(R) for
all subgroups H ≤ G.(iii) It is R-perfect, i.e. the underlying
complex ResG1 (X) ∈ Db(R) is perfect, and
it is H-cohomologically perfect, χH(X) = 0, for every Sylow
subgroup H ≤ G.(iv) It is E-cohomologically perfect for every
elementary abelian subgroup E ≤ G.(v) In D(R), we have XhH ∈
thick(
{RhK
∣∣K ≤ G}) for every subgroup H ≤ G.Proof. Implication (i)⇒(ii)
is Proposition 5.14. The implication (ii)⇒(iii) is
trivial(Definition 5.4). The equivalence between (iii) and (iv)
follows from Lemma 6.14,as explained in Remark 6.15. On the other
hand, assuming (iii), Lemma 6.14 tells
us that ResGH(X) ∈ P(H;R)\ for every Sylow subgroup H ≤ G. This
implies (i) byCorollary 3.19. So the four conditions
(i)⇔(ii)⇔(iii)⇔(iv) are equivalent.
Let us denote by J := thick({RhK
∣∣K ≤ G}) the thick subcategory of D(R) thatappears in (v). For
(v)⇒(ii), note that sing(RhK) = χK(R) = 0 by Corollary 5.15.So J ⊆
Ker(sing) and therefore XhH ∈ J implies χH(X) = sing(XhH) = 0.
Finally,for (i)⇒(v), since P(G;R)\ = Im(F̄ ) = thick(perm(G;R)) by
Theorem 3.21, itsuffices to prove that for every subgroups H,L ≤ G
we have (R(G/L))hH ∈ J. Thisfollows from the Mackey formula,
Proposition 5.10 and the definition of J. �
6.17. Remark. As in Remark 3.20, we can use Corollary 2.20 to
only test (iii),or (iv), for those p-subgroups of G with p
non-invertible on X (and in R).
6.18. Remark. Theorem 1.4 of the Introduction follows from
Theorems 3.21 and 6.16.
6.19. Remark. The inflation functor c∗ : D(R)→ D(RG) is monoidal
and its rightadjoint c! = (−)hG : D(RG) → D(R) is therefore lax
monoidal. In particular,c!c∗(1) = RhG is a ring object, namely the
‘cohomology ring’ of G with coefficientsin R, and every object X ∈
D(RG) gives rise to a module XhG over this ring.
With this in mind, and the fact that for every ring Λ we have
Dperf(Λ) =thick(Λ), the terminology ‘cohomologically perfect’ of
Definition 5.4 is somewhatjustified by the equivalent formulation
given in part (v) of Theorem 6.16.
-
22 PAUL BALMER AND MARTIN GALLAUER
The regular case is now trivial (compare Corollary 1.7 and
Scholium 3.15):
6.20. Corollary. Suppose that R is regular. Then P(G;R)\ =
Db(RG) and thecanonical functor Kb(RG)→ Db(RG) induces an
equivalence
(6.21)
(Kb(perm(G;R))
Kb,ac(perm(G;R))
)\∼−→ Db(RG).
Proof. We have P(G;R)\ = Db(RG) by Theorem 6.16 since all
obstructions χH(X)
trivially vanish in Dsing(R) = 0. We conclude by Theorem 3.21.
�
7. Localization of big categories
Neeman’s Theorem [Nee92] suggests that the equivalence (6.21) in
Corollary 6.20might be the compact tip of an iceberg, which indeed
emerges in this section.
7.1. Definition. The big homotopy category K(Mod(RG)) of
RG-modules admitssmall coproducts. We define the (big) derived
category of permutation modules
DPerm(G;R) := Loc(perm(G;R)) = Loc({R(G/H)
∣∣H ≤ G})as the localizing subcategory of K(Mod(RG)) generated
by permutation modules.
Since every generator R(G/H) is compact in K(Mod(RG)), we know
by [Nee92]that the triangulated category DPerm(G;R) is compactly
generated and its sub-category of compact objects is the bounded
homotopy category of \-permutationmodules (see Recollection
1.11):
(7.2) DPerm(G;R)c = thick(perm(G;R)) = Kb(perm(G;R)\).
7.3. Remark. This ad hoc definition does not do justice to the
derived category ofpermutation modules but it has the advantage of
simplicity and catches the rightcompacts. Let us now give a more
conceptual approach, justifying the name.
7.4. Notation. Let Perm(G;R) ⊆ Mod(RG) be the category of all
permutation RG-modules R(A) for possibly infinite G-sets A. It is
the closure of perm(G;R) undercoproducts. We have inclusions
DPerm(G;R) ⊆ K(Perm(G;R)) ⊆ K(Mod(RG)).Define the class QIG of
G-quasi-isomorphisms in K(Perm(G;R)) by
QIG =
{s : X → Y in K(Perm(G;R))
∣∣∣∣ sH : XH → Y H is aquasi-isomorphism for all H ≤ G}.
7.5. Proposition. The category DPerm(G;R) is obtained from the
homotopy cate-gory of all permutation modules by inverting
G-quasi-isomorphisms: The
compositeDPerm(G;R)�K(Perm(G;R))�K(Perm(G;R))[QI−1G ] is an
equivalence.
Proof. Since D := DPerm(G;R) is generated by the compacts R(G/H)
of K :=K(Perm(G;R)), it follows by Brown-Neeman Representability
that the inclusionD�K admits a right adjoint and the composite
D�K�K/D⊥ is an equivalence.Here we have D⊥ =
{cone(s)
∣∣ s ∈ QIG } since for all Z ∈ K, H ≤ G and n ∈ ZHomK(R(G/H)[n],
Z) ∼= HomK(Mod(R))(R[n], ZH) ∼= Hn(ZH). �
7.6. Remark. The above quotient functor K�K[QI−1G ] realizes
DPerm(G;R) as aBousfield colocalization of K = K(Perm(G;R)). It is
also a Bousfield localization(i.e. that quotient also admits a
fully faithful right adjoint) because K(Perm(G;R))is itself
compactly generated. See details in the expository paper
[BG20].
-
PERMUTATION RESOLUTIONS AND COHOMOLOGICAL SINGULARITY 23
7.7. Remark. In [BG20] we show that DPerm(G;R) is equivalent to
the derivedcategory of cohomological R-linear Mackey functors on G.
After suitably extendingDefinition 7.1 to profinite groups, we also
explain in [BG20] how DPerm(G;R) isequivalent to the triangulated
category of Artin motives DAM(F;R) in the senseof Voevodsky, over a
base field F whose absolute Galois group is G.
Let us now return to the relation with K(Inj(RG)).
7.8. Definition. Define a localizing subcategory of K(Inj(RG))
as follows:
K Injperm(G;R) := Loc(Qρ(R(G/H)) | H ≤ G
).
Since the functor Qρ : D(RG)�K(Inj(RG)) of (4.2) identifies
Db(mod(G;R)) withthe compact objects of K(Inj(RG)), we see that K
Injperm(G;R) is a compactly
generated triangulated category, and its compact part is
equivalent to P(G;R)\:
(7.9) P(G;R)\ = thick(R(G/H) | H ≤ G) Qρ−−→∼= K
Injperm(G;R)c.
The two subcategories we defined so far are related by the
functor Jλ of (4.4).
7.10. Lemma. For every subgroup H ≤ G, we have Jλ(R(G/H)) ∼=
Qρ(R(G/H))in K(Inj(RG)). Consequently, Jλ
(DPerm(G;R)) ⊆ K Injperm(G;R).
Proof. The first claim follows from Lemma 4.5 (c) since R(G/H)
is clearly bounded.The second claim follows by Definitions 7.1 and
7.8 and cocontinuity of Jλ. �
7.11. Notation. In view of Lemma 7.10, the restriction of Jλ to
DPerm(G;R) yieldsa well-defined coproduct-preserving and
compact-preserving exact functor
(7.12) F+ := (Jλ)|DPerm(G;R) : DPerm(G;R)→ K Injperm(G;R).
This functor F+ extends beyond compacts the functor F of the
Introduction:
7.13. Proposition. The following diagram commutes up to
isomorphism
(7.14)
Kb(perm(G;R))\
F(see Cor. 3.17)��
DPerm(G;R)c // //
(F+)c��
DPerm(G;R)
F+ (see (7.12))��
P(G;R)\ ∼=
(7.9)//
��
��
K Injperm(G;R)c // //
����
K Injperm(G;R)����
Db(mod(RG))Qρ
∼=// K(Inj(RG))c // // K(Inj(RG)).
Proof. Simply Lemma 4.5 (c) again, since Kb(perm(G;R))\ ⊆
K−(Mod(RG)). �
7.15. Theorem. Let G be a finite group and R a commutative
noetherian ring. Thefunctor F+ : DPerm(G;R) → K Injperm(G;R) is a
finite localization with respectto Kb,ac(perm(G;R)), i.e. it
induces a well-defined equivalence
(7.16) F̄+ :DPerm(G;R)
Loc(Kb,ac(perm(G;R)))
∼→ K Injperm(G;R).
On compacts, this equivalence identifies with the equivalence F̄
of (3.22), via (7.9).
Proof. By Lemma 4.5 (b), for every bounded acyclic X ∈
Kb,ac(perm(G;R)) wehave F+(X) = Jλ(X) = 0. We deduce that F
+ descends to the quotient as in (7.16).This functor F̄+
preserves coproducts and compact objects hence it suffices to
showit is an equivalence on compact objects. As F̄+ agrees with F̄
on compacts byProposition 7.13, this equivalence on compacts holds
by Theorem 3.21. �
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24 PAUL BALMER AND MARTIN GALLAUER
7.17. Corollary. If R is regular then we have K Injperm(G;R) =
K(Inj(RG)) andthe functor F+ : DPerm(G;R) → K(Inj(RG)) is a finite
localization with respectto Kb,ac(perm(G;R)). On compacts, F
+ induces the localization of Corollary 6.20.
Proof. This follows from Theorem 7.15, using Corollary 6.20.
�
7.18. Remark. Assume that R = k is a field. We mention as a
curiosity that, aslocalizing subcategories of K(Mod(RG)), we have
the inclusion
(7.19) K(Inj(kG)) ⊆ DPerm(G; k)and the finite localization F+ is
simply the left adjoint to that inclusion. This isquite remarkable:
For example, it says that for each finitely generated kG-moduleM
with injective resolution IM , the latter may be constructed out of
permutationmodules using triangles, suspensions and (arbitrary)
coproducts. In fact, these IMare generators for K(Inj(kG)) so this
fact is equivalent to the inclusion (7.19).
To prove it, we may assume the k-linear dual M∗ belongs to P(G;
k) and choosea 1-free permutation resolution P (1)→M∗. As P (1) ∈
P(G; k) by Proposition 2.7,we can find a 2-free permutation
resolution P (2)→ P (1). Continuing like this andpassing to
k-linear duals we construct a sequence
(7.20) I(1)f(1)−−−→ I(2) f(2)−−−→ I(3) f(3)−−−→ · · ·
where each I(n) is a bounded complex of permutation modules with
I(n)i injectivefor all i ≥ −n, and each f(n) is a
quasi-isomorphism. Note that the filtered colimitIM = colimn I(n)
is a resolution of M , and in each degree i, the sequence
(7.21) I(1)i → I(2)i → · · ·eventually consists of injective
kG-modules. It follows that IM is an injective res-olution of M .
(Indeed, injective, projective and flat modules coincide, and
filteredcolimits of flat are flat.) The final two steps are to show
that Perm(G; k) is closedunder filtered colimits in Mod(kG), which
follows from the comparison with co-homological Mackey functors in
[BG20], and that DPerm(G; k) is closed undersequential colimits in
Ch(Perm(G; k)), analogously to [BK08, Theorem 13.3].
8. Density and Grothendieck group
We want to use Thomason’s classification of dense subcategories
to derive con-sequences from the results of Sections 2 and 3.
Although some generalities workfor any regular ring R, we are
primarily interested in the field case.
8.1. Hypothesis. In this section, k is a field of characteristic
p > 0.
8.2. Recollection. Given an essentially small triangulated
category T we may con-sider its Grothendieck group, K0(T), the free
abelian group generated by isomor-phism classes of objects in T
quotiented by the relation [X] + [Z] = [Y ] for eachexact triangle
X → Y → Z → ΣX in T. In particular −[X] = [ΣX].
For each dense triangulated subcategory A ⊆ T (Recollection
2.11) the mapK0(A)→ K0(T) is injective ([Tho97, Corollary 2.3])
hence K0(A) defines a subgroupof K0(T). Conversely, each subgroup L
⊆ K0(T) defines a dense subcategory
A(L) :={X ∈ T
∣∣ [X] ∈ L in K0(T)}.By [Tho97, Theorem 2.1] these constructions
yield a well-defined bijection
{ dense triangulated subcategories of T } ∼←→ { subgroups of
K0(T) }.
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PERMUTATION RESOLUTIONS AND COHOMOLOGICAL SINGULARITY 25
8.3. Remark. We want to apply Thomason’s Theorem to the
triangulated subcat-egories P(G; k) and Q(G; k) of T = Db(kG)
introduced in Definition 2.8, whichare dense by Corollary 3.11. The
Grothendieck group of Db(kG) as a triangulatedcategory coincides
with the Grothendieck group of mod(kG) as an abelian category
K0(Db(mod(kG))) ∼= K0(mod(kG)) = G0(kG).
This Grothendieck group is free abelian on the set of
isomorphism classes of simplekG-modules. In the classic reference
[Ser77], Serre writes Rk(G) for G0(kG) andPk(G) for the
Grothendieck group K0(kG) of projective modules. We do not
adoptthis notation to avoid confusion with our coefficient ring R
and the category P(G; k).
Let us gather in one statement all the subgroups of G0(kG) that
we consider.
8.4. Proposition. We have the inclusions of subgroups in G0(kG)
= K0(mod(kG))
Z · [kG] ⊆
⊆
K0(kG) = K0(proj(kG))
⊆
KP0 (G; k) := K0(P(G; k)) ⊆ KQ0 (G; k) := K0(Q(G; k)) ⊆
G0(kG).
The quotients G0(kG)/K0(kG) and G0(kG)/KQ0 (G; k) are finite.
Specifically, these
quotients are finite abelian p-groups, whose exponent is a power
of p dividing |G|.
Proof. Injectivity of KP0 (G; k) → G0(kG) and KQ0 (G; k) →
G0(kG) follows from
density (Corollary 3.11) and Recollection 8.2. All inclusions in
the statement arethen straightforward, already for underlying
categories. The finite index of thesubgroup K0(kG), and therefore
of K
Q0 (G; k), follows from [Ser77, § 16, Theorem 35]:
There exists a power pn dividing |G| such that pn ·G0(kG) ⊆
K0(kG). (1) �
8.5. Example. The cokernel of the ‘Cartan’ homomorphism K0(RG)→
G0(RG) isnot always of finite exponent when R is not a field, even
for a DVR. Take R = Z(2)and G = C2 cyclic of order 2. Then K0(RC2)
= Z · [RC2] since RC2 is local. LetR+ = R with trivial C2-action.
Rationally, in G0(QC2) ∼= Z · [Q+] ⊕ Z · [Q−] forQ+ trivial and Q−
= Q with sign action, our [R+] maps to [Q+] but [RC2] mapsto [Q+] +
[Q−]. So no non-zero multiple of [R+] ∈ G0(RC2) belongs to
K0(RC2).
Applying Thomason’s classification (Recollection 8.2) we get for
instance:
8.6. Corollary. An M ∈ mod(kG) admits m-free permutation
resolutions for allm ≥ 0 if and only if its class [M ] ∈ G0(kG)
belongs to the subgroup KP0 (G; k).(Same for m-projective
p-permutation resolutions, with KQ0 (G; k) instead.) �
We do not have a description of KP0 (G; k) in general but it is
already remarkableto have a condition in terms of the class of M in
the Grothendieck group. Usingonly that free modules belong to P(G;
k) we get some interesting consequences.
8.7. Corollary. Let M ∈ mod(kG) and consider Ω(M) = Ker(kG ⊗k
M�M).Then M⊕Ω(M) admits m-free permutation resolutions for all m ≥
0. In particular,M ⊕ Ω(M) admits a finite resolution by finitely
generated permutation modules.
Proof. We have in G0(kG) that [M⊕Ω(M)] = [kG⊗kM ] ∈ KP0 (G; k)
since kG⊗kMis free. So we apply Corollary 8.6. The second part
follows by Corollary 2.5. �
1 The general assumptions of [Ser77, p. 115] hold for any k by
[Hoc17, Theorem, p. 23].
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26 PAUL BALMER AND MARTIN GALLAUER
8.8. Remark. Unlike KQ0 (G; k), the subgroup KP0 (G; k) ⊆ G0(kG)
is not of finite
index in general, simply because permutation modules are defined
integrally. Sothe subgroup KP0 (G; k) is always contained in the
image of G0(FpG) inside G0(kG)that can have infinite index when kG
has simple modules not defined over Fp.
8.9. Remark. The essential images of the canonical functors
Kb(perm(G; k)) →Db(kG) and Kb(perm(G; k)
\)→ Db(kG) are also dense triangulated subcategories.Indeed,
these essential images are the same as those of the functors
Kb(perm(G; k))
Kb,ac(perm(G; k))→ Db(kG) and
Kb(perm(G; k)\)
Kb,ac(perm(G; k)\)→ Db(kG).
As these functors are full by Theorem 3.21, their images are
triangulated subcat-egories. As these images contain P(G; k), they
are dense by Corollary 3.11. Infact, the right-hand functor is
already essentially surjective, as we shall see in The-orem 8.17.
By Thomason, it suffices to understand what happens on K0.
8.10. Proposition (Boltje/Bouc). The canonical homomorphism
K0(perm(G; k)\)→ G0(kG)
from the additive Grothendieck group of p-permutation modules
(a. k. a. the p-permu-tation ring [BT10], or trivial source ring
[Bol98]) is a surjection onto G0(kG). (
2)
Proof. Brauer’s Theorem in the modular case [Ser77, § 17.2]
asserts thatInd : ⊕H G0(kH)�G0(kG)
is surjective, where H runs through the so-called ΓK-elementary
subgroups of G(with notation of [Ser77, § 12.4]). So it suffices to
prove the result for G of thattype. In that case, we prove that
every simple kG-module M is p-permutation.
In the ‘easy case’ where G = C o Q with C (cyclic) of order a
power of p andQ of order prime to p, we can consider the non-zero
submodule MC of M . As Cis normal in G, it follows that MC is a
kG-submodule of M , hence equal to it. Inshort, M has trivial
restriction to the p-Sylow C of G, hence is p-permutation.
The ‘tricky case’ is when G = C o P where P is a p-Sylow and C
is cyclic oforder m prime to p. Using induction on |G| as in the
proof of [Ser77, § 17.3, Theo-rem 41], we reduce to the case where
M is a finite extension k′ = k[X]/f(X) wheref is an irreducible
factor of Xm − 1, on which P acts through k-automorphisms ofthe
field k′. As m and p are coprime, the cyclotomic extension k′/k is
separableand hence Galois. It follows from the normal basis theorem
that the k[P ]-modulek′ is permutation, and we conclude as before
(see Recollection 1.11). �
8.11. Remark. If we assume the field k ‘sufficiently large’ (cf.
[Ser77, p. 115]), e.g.algebraically closed, the above ΓK-elementary
subgroups are q-elementary for aprime q, i.e. of the form C×Q for a
q-group Q and a cyclic group C of order primeto q. In that case,
the p-Sylow of G is normal and we can apply the ‘easy case’ ofthe
above proof. The ‘tricky case’ was communicated to us by Serge
Bouc.
8.12. Remark. Robert Boltje gave us a different argument to
remove ‘k sufficientlylarge’ in Proposition 8.10, building a
natural section of K0(perm(G; k)
\)→ G0(kG).This uses the canonical induction formula for the
Brauer character ring, as well asGalois descent to reduce to the
case of k sufficiently large discussed above.
2 This homomorphism is rarely injective: Already G = Cp, we get
Z[x]/(x2 − px)→ Z.
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PERMUTATION RESOLUTIONS AND COHOMOLOGICAL SINGULARITY 27
In the remainder of this section, we are going to prove that in
fact Q(G; k) =Db(kG). Let us start with an easy observation
([Dre75, Theorem 1]).
8.13. Lemma. Let M be a finitely generated kG-module that admits
a k-basis Bwith the property that G · B ⊆ B ∪ (−B), that is, for
every g ∈ G and b ∈ B wehave g b ∈ B or −g b ∈ B. Then M is a
p-permutation module.Proof. If char(k) = p = 2 then M is
permutation. So assume p odd. It suffices toshow that M restricts
to a permutation module over a p-Sylow. So we can assumethat G is a
p-group and show in that case that M is also permutation.
Let m = dimk(M). Embed the elementary abelian 2-group (C2)m ↪→
Glm(k) as
(±1)m diagonally and the symmetric group Sm ↪→ Glm(k) as
permutation matricesand let Γ = (C2)
m ·Sm ≤ Glm(k) the subgroup of matrices that have exactly one±1
entry in each row and each column and all other entries zero. One
shows easilythat Γ = (C2)
moSm and in any case [Γ : Sm] = 2m. The basis B in the
statementyields a group homomorphism f : G→ Glm(k) that lands
inside Γ by hypothesis.
Now f(G) is a p-subgroup of Γ, hence contained in a p-Sylow.
Since [Γ : Sm] isprime to p, we can assume that up to Γ-conjugation
f(G) is contained in Sm. Thismeans that up to reordering the basis
and changing some signs, we can assume thatG acts on B via the
action of Sm on k
m, that is, by permuting the basis. �
8.14. Example. Let A be a finite G-set with n elements and let V
= k(A) thecorresponding permutation module. Let 0 ≤ s ≤ n and ΛsV
the s-th exteriorpower of V over k. It has dimension
(ns
)and it inherits a ‘diagonal’ G-action as a
quotient of V ⊗s, that is, such that g · (v1 ∧ · · · ∧ vs) =
(gv1) ∧ · · · ∧ (gvs).Now if we order the elements of A as A = {a1,
. . . , an} then ΛsV has the usual
basis B ={ai1 ∧ · · · ∧ ais
∣∣ 1 ≤ i1 < · · · < is ≤ n}. This basis satisfies
theassumptions of Lemma 8.13. Indeed (gai1)∧· · ·∧ (gais) is equal
to an element of Bup to the sign of the permutation of s letters
that rearranges gai1 , · · · , gais ∈ A inincreasing order. Hence
ΛsV is a p-permutation module.
Choose an order on the elements of G = {g1, . . . , gn}.8.15.
Recollection. (We follow the conventions of [Sta20, Tag 0621] for
Koszul com-plexes.) Let Kos(G; k) be the Koszul complex of k-vector
spaces associated with theaugmentation morphism � : kG → k
considered as a morphism of k-vector spacesk⊕n → k which is the
identity on each summand. Thus explicitly, Kos(G; k) isconcentrated
in degrees n, . . . , 0, and in degree s is the exterior product
Λsk(kG) asin Example 8.14 for A = G. Moreover, the differential
Kos(G; k)s → Kos(G; k)s−1sends a basis element gi1 ∧ · · · ∧ gis to
the alternating sum
s∑j=1
(−1)j−1gi1 ∧ · · · ∧ ĝij ∧ · · · ∧ gis
where the factor ĝij is removed.
8.16. Lemma. With the canonical ‘diagonal’ G-action in each
degree (Example 8.14),the complex Kos(G; k) is an acyclic complex
of kG-modules.
Proof. Acyclic is standard (over k), see for instance [Sta20,
Tag 0663]. For G-linearity of the differentials we compute for g ∈
G:
g ·s∑j=1
(−1)j−1gi1 ∧ · · · ∧ ĝij ∧ · · · ∧ gis =s∑j=1
(−1)j−1ggi1 ∧ · · · ∧ ĝgij ∧ · · · ∧ ggis
https://stacks.math.columbia.edu/tag/0621https://stacks.math.columbia.edu/tag/0663
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28 PAUL BALMER AND MARTIN GALLAUER
which is the image of ggi1 ∧ · · · ∧ ggis = g(gi1 ∧ · · · ∧ gis)
under the differential. �
8.17. Theorem. Let G be a finite group, and k a field of
characteristic p > 0. ThenQ(G; k) = Db(kG). In particular, every
kG-module admits a finite p-permutationresolution, and the
canonical functor (see Recollection 1.11)
Kb(perm(G; k)\)
Kb,ac(perm(G; k)\)
∼→ Db(kG)
is an equivalence, i.e. the left-hand quotient is already
idempotent-complete.
Proof. By the argument in Remark 8.9, all statements will follow
once we show thatKQ0 (G; k) = G0(kG). First, we claim that the
subgroup K
Q0 (G; k) ⊆ G0(kG) is an
ideal. Indeed, by Proposition 8.10, it suffices to show that KQ0
(G; k) is closed undermultiplying by the class of a p-permutation
module, which is straightforward.
We are therefore reduced to show that 1 = [k] belongs to KQ0 (G;
k). Forthis consider the Koszul complex Kos(G; k) of Recollection
8.15, and set K :=Kos(G; k)≥1[−1]. By Lemma 8.16, K is a resolution
of k by kG-modules. In degree0 we have K0 = kG which is free. In
degree s + 1 > 0 we have Ks+1 = Λ
sk(kG)
which is p-permutation by Lemma 8.13. This is enough by
Proposition 2.22. �
8.18. Corollary. Let G be a p-group, where p = char(k) > 0.
Then every finitelygenerated kG-module has a finite resolution by
permutation kG-modules. �
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Paul Balmer, UCLA Mathematics Department, Los Angeles, CA
90095-1555, USAEmail address: [email protected]
URL: https://www.math.ucla.edu/~balmer
Martin Gallauer, Oxford Mathematical Institute, Oxford, OX2 6GG,
UKEmail address: [email protected]
URL: https://people.maths.ox.ac.uk/gallauer
https://stacks.math.columbia.edu
1. Introduction2. Bounded permutation resolutions3. Essential
image of the functor F4. Big singularity category5. Cohomological
singularity6. Main result7. Localization of big categories8.
Density and Grothendieck groupReferences